Question: Derive the multiplier formula when all components are included in the aggregate demand equation. Explain your steps.

Expert answer

 

In order to derive the multiplier formula when all components are included in the aggregate demand equation, we must first recall the definition of the multiplier. The multiplier is defined as the change in output (or GDP) divided by the change in spending that caused it. In other words, the multiplier tells us how much output (or GDP) will increase for every dollar increase in spending.

 

Now that we have a refresher on the definition of the multiplier, let's take a look at the equation for aggregate demand. The equation for aggregate demand is:

 

AD = C + I + G + (X-M)

 

where:

 

AD = Aggregate Demand

 

C = Consumption Expenditures

 

I = Investment Expenditures

 

G = Government Expenditures

 

X = Exports

 

M = Imports

 

Now that we have the equation for aggregate demand, we can plug in the definition of the multiplier and solve for it. We know that:

 

Multiplier = Change in output / Change in spending

 

and we also know that:

 

AD = C + I + G + (X-M)

 

Therefore:

 

Multiplier = Change in output / Change in AD

= Change in output / (Change in C + Change in I + Change in G + (Change in X - Change in M)) … solving for the multiplier

 

𝐶+𝐼+𝐺+(𝑋−𝑀)=𝐴𝐷

 

We can see that the multiplier is equal to the inverse of the sum of the changes in all of the components of aggregate demand. In other words, the larger the sum of the changes in the components of aggregate demand, the smaller the multiplier will be.

 

Now that we have derived the multiplier formula when all components are included in the aggregate demand equation, let's take a look at an example.

 

Example:

 

Suppose that consumption expenditures increase by $100, investment expenditures increase by $200, government expenditures increase by $300, exports increase by $400, and imports decrease by $500. What is the value of the multiplier?

 

In order to solve this problem, we need to plug the given values into our equation for the multiplier. We know that:

 

Multiplier = 1 / (Change in C + Change in I + Change in G + (Change in X - Change in M))

 

and we also know that:

 

C = Consumption Expenditures

I = Investment Expenditures

G = Government Expenditures

X = Exports

M = Imports

 

Therefore, we can plug in the given values and solve for the multiplier. We get:

 

Multiplier = 1 / (100 + 200 + 300 + (400 - 500))

= 1 / (100 + 200 + 300 + (-100))

= 1 / 0

= undefined

 

since the denominator is equal to zero. Therefore, we can conclude that the value of the multiplier in this example is undefined.